Machine learning comes in two varieties:
\[(X_1, y_1), (X_2, y_2), ..., (X_n, y_n)\]
Can we learn \(f\) by optimising \(\theta\) on training data?
| country | designation | points | price | province | region_1 | region_2 | variety | winery |
|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux |
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
where \(\Sigma\) is dense.
\[\begin{equation} (y_1,y_2)' \sim \mathcal{N}(0, D) \end{equation}\]
where \(D\) is diagonal.
Remember, we’ve assumed:
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
\[\begin{equation} (\tilde x_{1,i}, \tilde x_{2,i}) = (x_{1,i}, x_{2,i}) - (\bar x_{1}, \bar x_{2}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = \frac{1}{n}\sum (\tilde x_{1,i}, \tilde x_{2,i})' (\tilde x_{1,i}, \tilde x_{2,i}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = P D P' \end{equation}\]
\[\begin{align} y_1 &= P_1' . (\tilde x_1, \tilde x_2)\\ y_2 &= P_2' . (\tilde x_1, \tilde x_2) \end{align}\]
| country | designation | points | price | province | region_1 | region_2 | variety | winery | review_id |
|---|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz | 1 |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 2 | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley | 3 |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 4 |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | 5 | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | 6 | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | 7 | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 8 | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström | 9 |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm | 10 |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | 11 | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars | 12 |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars | 13 |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | 14 | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene | 15 |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström | 16 |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall | 17 |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | 18 | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | 19 | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell | 20 |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz | 21 |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | 22 | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | 23 | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort | 24 |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin | 25 |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | 26 | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | 27 | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 28 |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar | 29 |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah | 30 |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | 31 | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas | 32 |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | 33 | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | 34 | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty | 35 |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | 36 | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | 37 | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | 38 | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | 39 | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | 40 | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | 41 | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | 42 | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | 43 | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | 44 | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | 45 | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | 46 | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | 47 | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin | 48 |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | 49 | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | 50 | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | 51 | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | 52 | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | 53 | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | 54 | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | 55 | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille | 56 |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | 57 | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank | 58 |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste | 59 |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | 60 | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | 61 | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | 62 | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | 63 | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | 64 | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | 65 | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens | 66 |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | 67 | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate | 68 |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek | 69 |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards | 70 |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond | 71 |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | 72 | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | 73 | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | 74 | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | 75 | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit | 76 |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | 77 | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars | 78 |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | 79 | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | 80 | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights | 81 |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | 82 | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | 83 | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | 84 | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | 85 | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | 86 | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | 87 | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | 88 | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven | 89 |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan | 90 |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | 91 | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | 92 | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | 93 | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci | 94 |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | 95 | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | 96 | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | 97 | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | 98 | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | 99 | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux | 100 |
| price | points |
|---|---|
| 235 | 96 |
| 110 | 96 |
| 90 | 96 |
| 65 | 96 |
| 66 | 95 |
| 73 | 95 |
| 65 | 95 |
| 110 | 95 |
| 65 | 95 |
| 60 | 95 |
| 80 | 95 |
| 48 | 95 |
| 48 | 95 |
| 90 | 95 |
| 185 | 95 |
| 90 | 95 |
| 325 | 95 |
| 80 | 95 |
| 290 | 95 |
| 75 | 95 |
| 24 | 95 |
| 79 | 95 |
| 220 | 95 |
| 60 | 95 |
| 45 | 95 |
| 57 | 94 |
| 62 | 94 |
| 105 | 94 |
| 60 | 94 |
| 60 | 94 |
| 15 | 90 |
| 37 | 90 |
| NA | 90 |
| 22 | 90 |
| 42 | 90 |
| 135 | 90 |
| 60 | 90 |
| 29 | 90 |
| 23 | 90 |
| 29 | 90 |
| 17 | 90 |
| 26 | 90 |
| 55 | 90 |
| 39 | 90 |
| 69 | 90 |
| 30 | 90 |
| 90 | 90 |
| 60 | 90 |
| 50 | 90 |
| 40 | 90 |
| 100 | 90 |
| 68 | 90 |
| 42 | 90 |
| 28 | 90 |
| 18 | 90 |
| 69 | 90 |
| NA | 90 |
| 25 | 90 |
| 30 | 90 |
| 60 | 90 |
| 30 | 91 |
| 36 | 91 |
| 25 | 91 |
| 45 | 91 |
| 23 | 91 |
| 36 | 91 |
| 38 | 91 |
| 85 | 91 |
| 50 | 91 |
| 60 | 91 |
| 85 | 91 |
| 45 | 91 |
| NA | 91 |
| 19 | 91 |
| 15 | 91 |
| 54 | 91 |
| 85 | 91 |
| 38 | 91 |
| 28 | 91 |
| 75 | 91 |
| 42 | 91 |
| 25 | 91 |
| NA | 91 |
| 59 | 91 |
| 85 | 91 |
| 80 | 91 |
| 45 | 91 |
| 22 | 91 |
| 65 | 91 |
| 50 | 91 |
| 10 | 86 |
| 12 | 86 |
| 22 | 86 |
| 13 | 86 |
| 10 | 86 |
| 14 | 86 |
| 18 | 86 |
| 36 | 86 |
| 15 | 86 |
| 10 | 86 |
Essentially these methods try to learn the structure of the local manifolds in the data. (Also referred to as graph-based or embedding methods.)
\[\begin{equation} p_{j|i} = \frac{\exp(-|x_i-x_j|^2/2\sigma_i^2)}{\sum_{k\neq i} \exp(-|x_i-x_k|^2/2\sigma_i^2)} \end{equation}\]
where \(\sigma_i\) is different for each point: points in dense areas given smaller variance than those in sparse areas
Need a distance metric, so define:
\[\begin{equation} p_{ij} = \frac{1}{N} (p_{j|i} + p_{i|j}) \end{equation}\]
yielding a similarity matrix with elements \(p_{ij}\)
Uses Student-t distribution with one degree of freedom (the Cauchy):
\[\begin{equation} q_{ij} = \frac{1/(1+|z_i-z_j|^2)}{\sum_{k\neq i} 1/(1+|z_i-z_k|^2)} \end{equation}\]
We can calculate the Kullback-Leibler divergence (not a distance!) from \(p_{ij}\rightarrow q_{ij}\):
\[\begin{equation} \text{KL} = \sum_{i,j} p_{ij} \log \frac{p_{ij}}{q_{ij}} \end{equation}\]
Minimise via gradient descent using \(\frac{\partial \text{KL}}{\partial z_i}\)
embedding method like t-SNE, with similar steps:
Minimise cross-entropy assuming connection is described by Bernoulli dist:
\[\begin{equation} \sum_e w_h(e) \log \frac{w_h(e)}{w_l(e)} + (1 - w_h(e)) \log \frac{1-w_h(e)}{1-w_l(e)} \end{equation}\]
where \(w_h\) are high dimensional weights; \(w_l\) are low dimensional weights.
n_neighbours: the number of approximate nearest neighbours used to construct the high dimensional graph. Low values \(\implies\) local structure; high values \(\implies\) global structure but slower run timemin_dist: minimum distance between points in low dimensional space. High values results in a more dense embeddingmin_distmin_distidea:
data assumed to be generated by a mixture of Gaussians:
\[\begin{align} p(x_i) &= \sum_k p(x_i|z_i=k) p(z_i=k)\\ &= \sum_k \mathcal{N}(x_i|\mu_k, \Sigma_k) \pi_k \end{align}\]
where \(\mu_k\) and \(\Sigma_k\) are the mean and covariance matrix of the Gaussian mode corresponding to cluster \(k\); \(\pi_k\) is the mixture proportion
repeat steps 1 and 2 repeatedly to draw a sample